On Nonlinear Complexity and Shannon’s Entropy of Finite Length Random Sequences

Lingfeng Liu,Bocheng Liu,Suoxia Miao

doi:10.3390/e17041936

Lingfeng Liu, Bocheng Liu + Show 1 more

Open Access

https://doi.org/10.3390/e17041936

Copy DOI

Journal: Entropy	Publication Date: Apr 1, 2015
Citations: 31	License type: CC BY 4.0

Affiliation: Nanchang University, Nanchang Institute of Technology

Abstract

Pseudorandom binary sequences have important uses in many fields, such as spread spectrum communications, statistical sampling and cryptography. There are two kinds of method in evaluating the properties of sequences, one is based on the probability measure, and the other is based on the deterministic complexity measures. However, the relationship between these two methods still remains an interesting open problem. In this paper, we mainly focus on the widely used nonlinear complexity of random sequences, study on its distribution, expectation and variance of memoryless sources. Furthermore, the relationship between nonlinear complexity and Shannon’s entropy is also established here. The results show that the Shannon’s entropy is strictly monotonically decreased with nonlinear complexity.

Highlights

Pseudorandom binary sequences have important uses in many fields, such as error control coding, spread spectrum communications, statistical sampling and cryptography [1,2,3]
Entropy 2015, 17 for generating pseudorandom bit sequences are based on the mid-square method, the linear congruential method, linear and nonlinear feedback shift registers, etc
Beirami et al indicated that the entropy rate plays a key role in the performance and robustness of chaotic map truly random number generators [14], and provided converse and achievable bounds on the binary metric entropy [15], which is the highest rate at which information can be extracted from any given map using the optimal bit-generation function, and et al Besides using the probability measure to evaluate a random sequence, many researchers provided the so-called deterministic complexity measures

Summary

Introduction

Pseudorandom binary sequences have important uses in many fields, such as error control coding, spread spectrum communications, statistical sampling and cryptography [1,2,3]. Entropy 2015, 17 for generating pseudorandom bit sequences are based on the mid-square method, the linear congruential method, linear and nonlinear feedback shift registers, etc. Beirami et al indicated that the entropy rate plays a key role in the performance and robustness of chaotic map truly random number generators [14], and provided converse and achievable bounds on the binary metric entropy [15], which is the highest rate at which information can be extracted from any given map using the optimal bit-generation function, and et al. Besides using the probability measure to evaluate a random sequence, many researchers provided the so-called deterministic complexity measures.

Nonlinear Complexity of Random Binary Sequences

The Relationship between Nonlinear Complexity and Shannon’s Entropy

Conclusions